Lemma 108.7.1. There exists a local ring $R$ and a maximal ideal $\mathfrak m$ such that the completion $R^\wedge $ of $R$ with respect to $\mathfrak m$ has the following properties
$R^\wedge $ is local, but its maximal ideal is not equal to $\mathfrak m R^\wedge $,
$R^\wedge $ is not a complete local ring, and
$R^\wedge $ is not $\mathfrak m$-adically complete as an $R$-module.